Ridge-penalised spectral least-squares estimation for point processes

Abstract

Penalised estimation methods for point processes usually rely on a large amount of independent repetitions for cross-validation purposes. However, in the case of a single realisation of the process, existing cross-validation methods may be impractical depending on the chosen model. To overcome this issue, this paper presents a Ridge-penalised spectral least-squares estimation method for second-order stationary point processes. This is achieved through two novel approaches: a p-thinning-based cross-validation method to tune the penalisation parameter, relying on the spectral representation of the process; and the introduction of a spectral least-squares contrast based around the asymptotic properties of the periodogram of the sample. The proposed method is then illustrated by a simulation study on linear Hawkes processes in the context of parametric estimation, highlighting its performances against more traditional approaches, specifically when working with short observation windows.